Assume that for coprime A, B and C and 1°) A^n=C^n-B^n [=(C-B)P], where n>2 and C>B>0 and1a°) P=p^n=C^{n-1}+C^{n-2}B+? +CB^{n-2}+B^{n-1}.

The proof of the FLT

If B=C-1 (obviously in this case, equation 1° has no solutions), we have:2°) C^n-(C-1)^n=1*[C^{n-1}+C^{n-2}(C-1)+? +C(C-1)^{n-2}+(C-1)^{n-1}] [=max P].

To get any other meaning of the number P, the number C-1 should be monotonically increased ? to judge from the left side of 2°, and THAT SAME number C-1 should monotonically decrease (up to the value B) ? judging by the right of the 2°.